Uniform Convergence of Schrödinger Cocycles over Simple Toeplitz Subshift

For locally constant cocycles defined on an aperiodic subshift, Damanik and Lenz (Duke Math J 133(1): 95-123, 2006) proved that if the subshift satisfies a certain condition (B), then the cocycle is uniform. In this paper, we study simple Toeplitz subshifts. We give a criterion that simple Toeplitz subshifts satisfy condition (B), and also give some sufficient conditions that they do not satisfy condition (B). However, we can still prove the uniformity of Schrödinger cocycles over any simple Toeplitz subshift. As a consequence, the related Schrödinger operators have Cantor spectrum of Lebesgue measure 0. We also exhibit a fine structure for the spectrum, and this helps us to prove purely singular continuous spectrum for a large class of simple Toeplitz potentials.